5th Australian Mathematical Olympiad Problems 1984

A1.  Show that there are no integers m, n such that 3 n4 - m4 = 131. A2.  ABC is equilateral. P and Q are points on BC such that BP = PQ = QC = BC/3. K is the semicircle on BC as diameter on the opposite side to A. The rays AP and AQ meet K at X and Y. Show that the arcs BX, XY and YX are all equal. A3.  The quartic x4 + (2a+1) x3 + (a-1)2x2 + bx + 4 factorises